Consider a uniform wire of mass $$M$$ and length $$L$$. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :

A wire of mass $$M$$ and length $$L$$ is bent into a semicircle. The relationship between the length and the radius is $$L = \pi R$$, giving $$R = \frac{L}{\pi}$$.

For a semicircular wire, every element of mass $$dm$$ lies at the same distance $$R$$ from the centre. Therefore, the moment of inertia about an axis perpendicular to the plane passing through the centre is $$I = MR^2$$.

Substituting $$R = \frac{L}{\pi}$$: $$I = M\left(\frac{L}{\pi}\right)^2 = \frac{ML^2}{\pi^2}$$.